Find the group of symmetries of the Cube.
The elements are:
$3$ rotations (by $\pi/2$ or $\pi$) about the centers of $3$ pairs of opposite faces.
$1$ rotation (by $\pi$) about the centers of $6$ pairs of opposite edges.
$2$ rotations (by $2\pi/3$) about $4$ pairs of opposite vertices (diagonals).
Together with the identity this accounts for all $24$ elements (the order of the group of direct symmetries is $24$).
Every rotation determines a permutation of the four diagonals and this defines the isomorphism, which tells us that the group of symmetries of the Cube is the group $S_4$.
Is my approach correct? This is an exercise I have to solve as part of an academic list of problems, so I want to make sure it's perfect and strict mathematically. If not, please tell me with details how it should be.